Preheat the oven to 350°F (175°C). Grease and flour an 8-inch (20 cm) square baking pan.
In a medium bowl, whisk together the all-purpose flour, baking soda, and salt. Set aside.
In a small bowl, combine the whole milk and white vinegar. Stir and let sit for 5 minutes to create a buttermilk substitute.
In a large bowl, beat the softened butter and granulated sugar together with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition. Mix in the vanilla extract.
Gradually add the flour mixture to the butter mixture alternately with the milk-vinegar mixture, beginning and ending with the flour mixture. Mix just until combined after each addition.
Fold in the mashed bananas gently until evenly incorporated.
Pour the batter into the prepared baking pan and smooth the top with a spatula.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and let the cake cool in the pan on a wire rack for 15 minutes.
Run a knife around the edges of the pan to loosen the cake, then invert onto a wire rack to cool completely before slicing and serving.